We carry out MHD simulations with the pluto
code [1]}, [2]}, [3]}.
The calculations are performed using a 2.5-dimensional spherical coordinate
system (\(O\) ;\(r\) ,\(\theta \) ,\(\phi \) ) of origin \(O\) , with \(r\)  the radial direction,
\(\theta \)  the poloidal coordinate and \(\phi \)  the toroidal component.
The coordinate system \([r_{\rm in},r_{\rm out}]\times [0,\pi ]\times [0,2\pi ]\) 
is mapped with a mesh that is uniform along the polar and toroidal directions,
while it expands logarithmically along the radial direction.
It permits to reach high spatial resolutions close to the stellar wind
boundary \(r_{\rm in}=0.02\, \rm pc\), while reducing the total
number of grid zones in the simulation domain and reducing the computational costs.
In total, we use \(200\times \,200\times \, 1\)  grid zones.
Outflow boundary conditions are assigned at the outer boundary
\(r_{\rm out}=10\, \rm pc\), reflective
boundary conditions are imposed along the
symmetry axis \(\theta =0\)  and \(\theta =\pi \) , while periodic boundary
conditions are used at the borders \(\phi =0\)  and \(\phi =2\pi \) .

Since the inner boundary is much larger than the radius of the photosphere
(\(r_{\rm in} \gg R_{\star }\) ), stellar gravity and wind acceleration mechanisms
such as coronal heating and radiation pressure, are neglected in the
computational domain. Therefore, the acceleration of the stellar wind to the
terminal velocity is not modelled in the simulations.

